SRC/sggesx.f(3)

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💖 SRC/sggesx.f 💖

okay bestie, welcome to the SRC/sggesx.f(3) page from the Library Functions Manual — this one is long, so hydrate and keep your emotional support water bottle close. ✨ The routine is sggesx, and she is doing the absolute most in the most iconic way. 💅

🌸 NAME 🌸

SRC/sggesx.f — that's the name section, no cap. 🫶

🌈 SYNOPSIS 🌈

🫶 Functions/Subroutines 🫶

subroutine sggesx — SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

Here's the callable, character-for-character, because we do not mess with signatures, periodt:

subroutine sggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
    ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde,
    rcondv, work, lwork, iwork, liwork, bwork, info)

🦋 Function/Subroutine Documentation 🦋

Full declaration, exactly as written, because signatures are sacred:

subroutine sggesx (character jobvsl, character jobvsr, character sort,
    external selctg, character sense, integer n, real, dimension( lda, * )
    a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer sdim,
    real, dimension( * ) alphar, real, dimension( * ) alphai, real,
    dimension( * ) beta, real, dimension( ldvsl, * ) vsl, integer ldvsl,
    real, dimension( ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 )
    rconde, real, dimension( 2 ) rcondv, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer liwork, logical,
    dimension( * ) bwork, integer info)

One-liner: SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

🎯 Purpose 🎯

okay so here's the vibe: SGGESX computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and — optionally — the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization:

(A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). 💫

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero. 🧚

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S happen by making the corresponding elements of T have the form:

[  a  0  ]
[  0  b  ]

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues. 🔥

🎀 Parameters 🎀

JOBVSL 💅

JOBVSL is CHARACTER*1.

= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.

JOBVSR 💅

JOBVSR is CHARACTER*1.

= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.

SORT 💅

SORT is CHARACTER*1. It specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELCTG).

SELCTG 💅

SELCTG is a LOGICAL FUNCTION of three REAL arguments. SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected.
Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3.

SENSE 💅

SENSE is CHARACTER*1. It determines which reciprocal condition numbers are computed.

= 'N': None are computed;
= 'E': Computed for average of selected eigenvalues only;
= 'V': Computed for selected deflating subspaces only;
= 'B': Computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. 🫶

N 💅

N is INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.

A 💅

A is REAL array, dimension (LDA, N).

On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur form S.

LDA 💅

LDA is INTEGER. The leading dimension of A. LDA >= max(1,N).

B 💅

B is REAL array, dimension (LDB, N).

On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur form T.

LDB 💅

LDB is INTEGER. The leading dimension of B. LDB >= max(1,N).

SDIM 💅

SDIM is INTEGER.

If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.)

ALPHAR 💅

ALPHAR is REAL array, dimension (N).

ALPHAI 💅

ALPHAI is REAL array, dimension (N).

BETA 💅

BETA is REAL array, dimension (N).

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). 😭

VSL 💅

VSL is REAL array, dimension (LDVSL,N).

If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.

LDVSL 💅

LDVSL is INTEGER. The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.

VSR 💅

VSR is REAL array, dimension (LDVSR,N).

If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.

LDVSR 💅

LDVSR is INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

RCONDE 💅

RCONDE is REAL array, dimension ( 2 ).

If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.

RCONDV 💅

RCONDV is REAL array, dimension ( 2 ).

If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition numbers for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.

WORK 💅

WORK is REAL array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK 💅

LWORK is INTEGER. The dimension of the array WORK.

If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else LWORK >= max( 8*N, 6*N+16 ).

Note that 2*SDIM*(N-SDIM) <= N*N/2.

Note also that an error is only returned if LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

IWORK 💅

IWORK is INTEGER array, dimension (MAX(1,LIWORK)).

On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

LIWORK — bestie, this one is an INTEGER, and it's literally the dimension of the array IWORK. If SENSE = 'N' or N = 0, then LIWORK >= 1; otherwise LIWORK >= N+6. And if LIWORK = -1, a workspace query is assumed: the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. No cap, that's the whole vibe.
BWORK — BWORK is a LOGICAL array, dimension (N). Not referenced if SORT = 'N'. So if sorting is off, this array is just chilling, unbothered. 💅
INFO — INFO is an INTEGER, and it's giving very much "read the room" energy:
- = 0: successful exit. Slay.
- < 0: if INFO = -i, the i-th argument had an illegal value. Not cute — somebody passed bad tea.
- = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
- > N:
  - =N+1: other than QZ iteration failed in SHGEQZ
  - =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
  - =N+3: reordering failed in STGSEN.

🌸 AUTHOR

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔍 FURTHER DETAILS

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / RCONDV( 2 ).

See LAPACK User's Guide, section 4.11 for more information.

Definition at line 361 of file sggesx.f.

🌸 AUTHOR (Doxygen's version)

Generated automatically by Doxygen for LAPACK from the source code.

💖 That's the whole serve, bestie: LAPACK, Version 3.12.0, page source SRC/sggesx.f(3). Bye and good luck out there! 💅✨

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